Primitive normal pairs of elements with one prescribed trace

This is the authors' abstract. We don't add key points for this paper.

Let q, n, m ∈ N such that q is a prime power, m ≥ 3 and a ∈ F. We establish a sufficient condition for the existence of a primitive normal pair ($α$, $f(α)$) in Fqᵐ over Fq such that Tr_Fqᵐ/Fq(α⁻¹)=a, where f(x) ∈ Fqᵐ(x) is a rational function with degree sum n. In particular, for q=5ᵏ, ~k ≥ 5 and degree sum $n=4$, we explicitly find at most 11 choices of $(q, m)$ where existence of such pairs is not guaranteed.